## Monday, 18 March 2019

### Sorry for my absence! AND a note on the abstract

Wow, it has been nearly a year since I have blogged. It doesn't feel that long, and yet I know it has been a long time. I have to apologise to anyone who has missed me (I can't imagine why you would), but the work involved in writing and then bringing a book to the point it can be published is quite something. Alongside this, I have been working hard on our curriculum developments. Before anyone asks, no this wasn't something levied on me by my school in response to Ofsted's new focus on curriculum. It was planned development that we instigated as a department, in response to the reading and development work we had done. I am really excited about its potential, but it has been quite a job of work. There have been new scheme documents to write, and new lesson materials to develop. New assessments to write, and new homework booklets to put together. I am planning a big launch of this at some point, but it won't be this academic year as so far we only have completed the materials for Year 7 and I want at least Year 8 done before we make it all public (plus I have to get permission from my school and team as well!) Hopefully it will be worth the wait. But in the meantime a more recent reflection.

A minor disagreement crossed my Twitter feed a couple of weeks back about the nature of the second term in 7 – 3y. The question was posed as to whether the second term is 3y or -3y. To answer this question I want to focus on what "7 – 3y" actually is.

The first thought is probably "its an (algebraic) expression". Totally correct. But still only words. What is 7 – 3y? This is where it gets difficult. A mathematical entity? A thing?

The truth is 7 – 3y is a pure concoction of thought. It is nothing except how it exists in our minds. That isn't to say that there aren't real phenomena that can be related to the expression. But they aren't the expression. The expression is just there, as an abstraction of the mind. And that means it can be whatever I want it to be. Or rather it is as I choose to make sense of it. If I understand that it can be seen as the difference between 7 and 3y, then I can choose to see it like that. If I can make sense of it as 7 and -3y then I am allowed to do that as well.

For me, this is why it is important for us to ensure we support our pupils in developing understanding. So much of maths only exists in the ways that we make sense of it. Even well established concepts such as addition only exist for us in the ways we are able to make sense of them. Addition may have arisen out of practical ideas, such as collecting objects together, but it has far surpassed that since it has been applied to things like irrational or complex numbers. It is now an abstract concept, there to be made of what I can. So long as I don't contradict the results of other ways of making sense I am fine (for example, I can't just decide that 3 + 5 is going to be 9 - any way I have to make sense of addition must result in 3 + 5 being 8).

If we don't support pupils to make sense of these concepts, to have different ways of seeing and manipulating abstract mathematics within its rules and established prior results, then our pupils will never be fluent in mathematics. They will not have a chance to attain the understanding of which they are heirs to. And then they won't get chance to delve into "the best that has been thought and said" in the field of mathematics.